Performance Analysis of UAV-IRS Relay Multi-Hop FSO/THz Link

Abstract

As the era of sixth-generation (6G) communications approaches, there will be an unprecedented increase in the number of wireless internet-connected devices and a sharp rise in mobile data traffic. Faced with the scarcity of spectrum resources in traditional communication networks and challenges such as rapidly establishing communications after disasters, this study leverages unmanned aerial vehicles (UAVs) to promote an integrated multi-hop communication system combining free-space optical (FSO) communication, terahertz (THz) technology, and intelligent reflecting surface (IRS). This innovative amalgamation capitalizes on the flexibility of UAVs, the deployability of IRS, and the complementary strengths of FSO and THz communications. We have developed a comprehensive channel model that includes the effects of atmospheric turbulence, attenuation, pointing errors, and angle-of-arrival (AOA) fluctuations. Furthermore, we have derived probability density functions (PDFs) and cumulative distribution functions (CDFs) for various switching techniques. Employing advanced methods such as Gaussian–Laguerre quadrature and the central limit theorem (CLT), we have calculated key performance indicators including the average outage probability, bit error rate (BER), and channel capacity. The numerical results demonstrate that IRS significantly enhances the performance of the UAV-based hybrid FSO/THz system. The research indicates that optimizing the number of IRS elements can substantially increase throughput and reliability while minimizing switching costs. Additionally, the multi-hop approach specifically addresses the line-of-sight (LoS) dependency limitations inherent in FSO and THz systems by utilizing UAVs as dynamic relay points. This strategy effectively bridges longer distances, overcoming physical and atmospheric obstacles, and ensures stable communication links even under adverse conditions. This study underscores that the enhanced multi-hop FSO/THz link is highly effective for emergency communications after disasters, addressing the challenge of scarce spectrum resources. By strategically deploying UAVs as relay points in a multi-hop configuration, the system achieves greater flexibility and resilience, making it highly suitable for critical communication scenarios where traditional networks might fail.

1. Introduction

It is projected that the arrival of the sixth-generation (6G) communications era will result in both an unparalleled rise in the quantity of internet devices connected wirelessly and a sharp growth in mobile data traffic [1]. This not only highlights a major obstacle for future communication networks’ transmission capacity, but it also shows how inadequate present communication technologies are to handle spectrum congestion [2,3], and resolve the problem of limited spectrum resources [4]. Addressing these challenges has become a focal point of research within the field of communications.

Due to their large bandwidth and fast wireless data transmission capabilities, free-space optical (FSO) and terahertz (THz) communication have garnered a lot of attention in the face of spectrum shortage and congestion problems. These technologies provide promising solutions to alleviate the current shortage of radio frequency (RF) communication bands [5,6]. The future generation of high-speed wireless communication systems will be supported by FSO in particular because of its many advantages, which include low implementation costs, high security, license-free spectrum use, large bandwidth, and ease of deployment [7]. Furthermore, FSO has demonstrated its distinct benefits in resolving the “last mile” issue with post-5G (B5G) communication systems [8]. However, in practical applications, FSO systems still face challenges such as atmospheric turbulence, mis-aimed shots, and interference from extreme weather conditions (such as rain and fog), which can lead to unreliable transmission links. Research and application of THz communication are particularly important to alleviate these problems that FSO encounters. THz communication shows great potential in high-data-rate propagation. Its frequency band is between millimeter waves and infrared light bands, which can provide high security, large directional gain, and wide operating bandwidth [3,9,10]. THz supports line-of-sight (LoS) and non-LoS (NLoS) propagation. Under conditions of haze and atmospheric turbulence, THz propagation performs better than FSO. However, in clear weather, the performance of THz communication may not be as good as FSO [11], mainly because of multipath fading caused by molecular interactions and energy absorption in the propagation medium [12]. To sum up, FSO and THz communications each have their own advantages and limitations. The strategy of parallel deployment of FSO and THz can not only give full play to the complementary advantages of the two technologies under different environmental conditions, but also provide an effective solution to the scarcity of spectrum resources and spectrum congestion problems faced by future wireless communication systems.

Nevertheless, both THz and FSO communications encounter significant challenges when it comes to real-world implementations [13,14], particularly with regard to severe path fading and channel degradation. To counteract the detrimental impacts of these factors, the adoption of relay technology has proven to be an efficacious method for enhancing the wireless transmission rate and broadening the scope of wireless coverage. Taking into account the presence of obstacles, the strategic placement of relays at optimal locations is critical in the system’s architecture. Moreover, given the high flexibility of unmanned aerial vehicles (UAVs), utilizing them as mobile relays represents an innovative and promising approach. Relative to conventional ground-based relay systems, UAV-mounted relays offer enhanced adaptability to identify the most favorable communication conditions and rapidly establish line-of-sight connections. In [15], the authors delve into modeling and evaluating the performance of a UAV-based FSO communication system, considering atmospheric turbulence, pointing inaccuracies, and angle-of-arrival (AOA) variations. Meanwhile, reference [16] provides a detailed analysis of the cumulative distribution function (CDF) and the probability density function (PDF) for a multi-hop UAV FSO communication system model, also under the influence of atmospheric turbulence, pointing errors, and AOA fluctuations. They derive explicit expressions for both the link outage probability and the end-to-end outage probability. Furthermore, they optimize the system’s performance by adjusting parameters such as beamwidth, field of view, and the UAV’s positioning. Reference [17] investigates a dual-hop hovering UAV FSO communication system that leverages decode-and-forward (DF) relaying, which takes into consideration numerous aspects that can affect the system’s performance, such as atmospheric route loss, atmospheric turbulence, aiming mistakes, and AOA fluctuations.

Although the studies mentioned above describe the progress made in using UAVs as mobile relays to overcome some of the limitations of conventional ground-based relays, they do not address the ability to achieve 360° panoramic coverage with full-angle reflections or ensure the attainment of the maximum signal-to-noise ratio (SNR) [18]. The emergence of intelligent reflecting surface (IRS) technology has laid a strong foundation for the development of advanced, intelligent, and controllable wireless communication systems that are expected to be part of future 6G networks [19]. IRS is composed of an array of electromagnetic material with numerous reflective elements, each capable of directionally reflecting incoming waves without necessitating an external power supply. Optimizing the number of reflective elements on an IRS, as well as adjusting its emission coefficients, makes it a cost-effective and easily deployable solution for enhanced communication systems. Reference [20] delved into the intricacies of an IRS-assisted hybrid RF/THz system, which utilizes a fixed-gain amplification forwarding protocol. They derived the CDF for the end-to-end channel gain of the hybrid RF/THz link, as well as the closed-form expressions for the PDF. Utilizing these statistical characteristics, they calculated the closed-form expressions for key performance metrics, including the outage probability, the average bit error rate (BER), and the average channel capacity. Additionally, they conducted asymptotic analyses to evaluate the system’s performance in terms of outage probability and average BER under extreme operating conditions. Reference [21] explores IRS-assisted RF and wireless optical communication systems, as well as the performance of wireless optical communication systems using the gamma–gamma distribution and the generalized Magara distribution. Reference [22] proposed a theoretical framework for cascaded IRS composite turbulence and pointing errors, provided a PDF and CDF of cascaded composite turbulence and pointing errors, from which new closure formulas were derived, and investigated cascaded multi-IRS FSO and THz wireless systems.

The insights gleaned from the aforementioned studies have spurred researchers to explore the integration of IRS with UAVs and IoT infrastructure to enhance overall system performance. Reference [23] studied a UAV-based IRS in a hybrid FSO/RF dual-hop communication system, with the FSO acting as a secondary link to the RF, which considered the effect of IRS reflective elements on phase shift errors and provided closed-form formulae for the average symbol BER and spectral efficiency of the system. Reference [24] explores the deployment of a legitimate UAV as a relay, alongside an IRS, to enhance the reach of free-space optical communication systems. Reference [25] develops a model for an IRS-assisted three-hop communication system, integrating RF and FSO for the initial two hops and underwater wireless optical communication (UWOC) for the final hop, formulating closed-form expressions for system performance metrics. Reference [26] investigates a hybrid backhaul network that incorporates FSO and THz technologies, augmented by reconfigurable IRS. However, the integration of UAVs as flexible relay nodes in this configuration remains unexplored.

Building upon the aforementioned studies, it is observed that the majority of research concentrates on dual-hop and triple-hop configurations, often employing conventional RF/FSO hybrid systems. However, there is a notable absence of integration of THz technology with IRS-UAV systems as an alternative backup for FSO links. Additionally, while evaluating multi-hop IRS-assisted systems, the scholarly focus predominantly lies on deriving closed-form expressions for system outage probabilities. This indicates a gap in the literature, as there is a dearth of comprehensive assessments encompassing various dimensions of system performance. Furthermore, several studies have overlooked the influence of the number of IRS reflector units, a critical parameter for system performance. Addressing this oversight, the current study introduces a novel communication system architecture that involves mounting an IRS onto a UAV. This design facilitates the creation of an IRS-UAV assisted multi-hop FSO/THz communication system. The multi-hop relay approach employed by this system not only broadens the communication system’s reach but also enhances the link quality by utilizing the THz band as a contingency for the FSO link. By implementing both hard-switching and soft-switching strategies, it amalgamates the strengths of both technologies to bolster communication quality. Moreover, the IRS-UAV support scheme augments system performance on multiple fronts, notably by amplifying the signal strength beyond that of conventional hybrid FSO/THz systems. Additionally, this study employs the decode-and-forward (DF) protocol and conducts a thorough evaluation of the system’s performance. It specifically investigates the UAV-based IRS-assisted hybrid multi-hop FSO/THz link system, wherein the FSO link is subject to gamma–gamma fading and THz links have pointing issues and follow a distribution.

The structure of the remainder of this paper is as follows: Section 2 delineates the system and channel models. Section 3 presents a detailed performance analysis of the hybrid multi-hop FSO/THz link system. Section 4 provides further numerical results. Section 5 offers concluding remarks.

2. System and Channel Models

Before analyzing this system, the following two assumptions need to be made:

• Assume that the UAVs are equipped with advanced stabilization technologies and GPS systems that provide high positional accuracy. This allows us to focus on the effects of environmental factors such as wind and turbulence under controlled conditions.
• Assume that the IRS can maintain a consistent performance level despite changes in the environment. This involves considering that the IRS components are robust enough to handle variations in temperature, humidity, and physical disruptions without significant degradation in performance.

The IRS-assisted multi-hop FSO/THz link system for UAVs is schematically diagrammed in Figure 1. It consists of an IRS-UAV relay node, a receiving user with a directional antenna, and a transmitting base station. To utilize their complementary benefits, this system combines FSO and THz links, with THz as the secondary link and FSO as the primary link. It improves communication efficiency and reliability by using both hard- and soft-switching strategies to ease the transition between FSO and THz networks.

Due to the absence of a direct LoS path between the transmitting base station and the receiving user, by equipping UAVs with a hypersurface-based IRS platform and using them as relay nodes, a virtual line-of-sight path can be created. We consider an IRS composed of programmable reflective elements, engineered to tailor communication system performance for specific scenarios. The transmitting base station is equipped with laser transmitters and an equal number of THz antennas, facilitating concurrent FSO and THz signal transmission. These signals are reflected by the IRS’s elements at each IRS-UAV relay node, traversing such nodes before reception by the user. The system employs intensity modulation with direct detection and opts for the BPSK modulation scheme. The data, modulated using BPSK, are conveyed over the FSO and THz links and, after relay through the IRS-UAV nodes, are directly detected by the receiving user.

Table 1. List of parameters.

Parameter	Definition
$P_{FSO\_i}$	Optical power transmitted at S
$P_{THz\_i}$	THz power transmitted at S
$\eta$	Photoelectric conversion factor
$S_i$	Transmitted signal
W	Number of antennas at S and D, number of IRS elements
N	Number of UAV
${\alpha }_i,{\beta }_i$	Shape parameters of the G-G fading
$A_{ij}$	Fraction of power collected at the focal point $r=0$
$g_{ij}^2$	Ratio of the radii of equivalent beamwidths
${\sigma }_{s\_ij}$	Variance of the pointing error
$L_{FSO\_i-1}$,$L_{THz\_i-1}$	(i − 1)th link distance
${\psi }_{AoA,i}$	Field of view (FoV) of the i-th link
$m_i$	Variable dependent on link type
${\sigma }_{ij,u}^2$	Displacement variance due to the positional offset of the i-th node
${\sigma }_{i-1j,q}^2$	Displacement variance due to the (i − 1)-th node direction offset
${\sigma }_{i-1j,angle}^2$	Displacement variance due to the (i − 1)-th node position offset
${\mu }_{THz\_i},{\alpha }_{THz\_i}$	Fading parameters of THz
${\widehat{h}}_f^{{\alpha }_{THz\_i}}$	$\alpha$-root mean value of fading channel envelope
${\gamma }_{th}^{FSO}$	SNR threshold of FSO link
${\gamma }_{th}^{THz}$	SNR threshold of THz link
${\gamma }_{th,u}^{FSO}$	Upper SNR threshold of FSO link
${\gamma }_{th,l}^{FSO}$	Lower SNR threshold of FSO link
$p,q$	Represent different binary modulations schemes

summarizes all the parameters used in the paper.

2.1. FSO Link

To enhance the received signal quality and system robustness, the transmitting base station S is outfitted with W laser transmitters, establishing W optical links for data transmission. Each relay node adjusts the direction and phase of incoming signals via its reflective elements, thereby ensuring efficient propagation to the subsequent node. The receiver employs W antennas, utilizing the maximum ratio combination (MRC) technique for optimal signal reception. The system encompasses $N+1$ links, with the base station as node 0 and the user terminal (D) as node $N+1$. The optical signal at the i-th node on the i-th link is mathematically represented as follows:

$$y_{FSO\_i}\left(t\right)=\left(P_{FSO\_i}\eta \sum _{i=1}^W{h}_{FSO\_i}\right)S_i+N_{FSO\_i},i=1\dots N+1,$$

(1)

In this framework, $P_{FSO\_i}$ signifies the optical power transmitted by the base station; $\eta$ represents the photoelectric conversion factor; $h_{FSO\_i}$ corresponds to the total channel gain of the i-th link, defined by $h_{FSO\_i}=h_{l,FSO\_i}{h}_{a,FSO\_i}{h}_{p,FSO\_i}{h}_{aoa,FSO\_i}$; $h_{l,FSO\_i}$ quantifies atmospheric loss; $h_{a,FSO\_i}$ measures atmospheric turbulence; $h_{p,FSO\_i}$ assesses pointing error; and $h_{aoa,FSO\_i}$ reflects link disruption caused by AOA fluctuations. Let $S_i$ denote the signal received from the preceding node, $N_{FSO\_i}$ represent the signal-independent additive Gaussian white noise characterized by a mean of 0 and a variance of ${\sigma }_0^2$, and $N_{FS{O}_{-}i}\sim \mathcal{N}(0,{\sigma }_0^2)$ and ${\gamma }_{\mathrm{FSO}\_\mathrm{i}}={\left(P_{FSO\_i}\eta {\Sigma }_{i=1}^W{h}_{FSO\_i}\right)}^2/{\sigma }_0^2$ signify the instantaneous SNR at the i-th node. Next, examine the four parameters mentioned above in terms of their statistical properties.

Derived from reference [27], the PDF for channel coefficients, influenced by pointing errors, atmospheric turbulence, and atmospheric attenuation, is obtained as follows:

$$\begin{array}{cc}\hfill & f_{{h_{FSO\_i}}^1}\left({h_{FSO\_i}}^1\right)={\displaystyle \frac{{\alpha }_i{\beta }_i{g}_{ij}^2}{A_{ij}{h}_{l,FSO\_i}\Gamma \left({\alpha }_i\right)\Gamma \left({\beta }_i\right)}}\times {\mathrm{G}}_{1,3}^{3,0}\left({\displaystyle \frac{{\alpha }_i{\beta }_i{h_{FSO\_i}}^1}{A_{ij}{h}_{l,FSO\_i}}}\left|\begin{array}{c}-;g_{ij}^2\\ g_{ij}^2-1,{\alpha }_i-1,{\beta }_i-1;-\end{array}\right.\right),\hfill \end{array}$$

(2)

where the parameter $g_{ij}^2={w_{eq\_ij}}^2/4{{\sigma }_{s\_ij}}^2$ denotes the ratio of the radii of equivalent beamwidths at the receiver (RX), while $A_{ij}$ signifies the fraction of power collected at the focal point $r=0$, calculable via parameter $u=\sqrt{\pi }{a}_{ij}/\sqrt{\pi }{w}_{d\_ij}$. The term r is defined as the pointing error, quantified by the radial distance between the centers of the transmitted and received beams. The variance of the pointing error displacement at the RX is indicated by ${\sigma }_{s\_ij}$, with a circular detector beam characterized by a radius $a_{ij}$, The maximum radius at a given distance d on the i-th link is denoted by $w_{d\_ij}$ and $w_{eq\_ij}$ represents the equivalent beam width, calculated as ${w_{eq\_ij}}^2={w_{d\_ij}}^2\times \sqrt{\pi }·erf\left(u\right)/2u·\exp (-u^2)$. $g_{ij}^2$ is influenced by the IRS-UAV link type and node fluctuation. G(·) is the Meijer’s G function; because we installed the IRS on the UAV, the IRS-UAV relay node is analyzed as a whole for determining the aiming inaccuracy. Furthermore, the link types in this system can be split into three categories, S-IRS-UAV(SU), IRS-UAV-IRS-UAV(UU), and IRS-UAV-D(UD), and the pointing error parameters of different types of connections differ, and we assume the pointing error parameters of the same types of links to be the same. According to references [15,16], the specific formula for parameter ${{\sigma }_{s\_ij}}^2$ is provided in Equation (3) and the turbulent parameters for large and small scales are denoted by ${\alpha }_{\mathrm{i}}$ and ${\beta }_i$, respectively, as detailed in Equation (3):

$$\begin{array}{cc}\hfill {\sigma }_{s\_ij}^2& =\left\{\begin{array}{cc}{\sigma }_{ij,u}^2+{\sigma }_{i-1j,q}^2\hfill & \mathrm{if}i=1\hfill \\ 2{\sigma }_{ij,u}^2+L_{\mathit{FSO}\_i-1}{\sigma }_{i-1j,angle}^2\hfill & \mathrm{if}i=2,\dots ,N\hfill \\ {\sigma }_{ij,u}^2+{\sigma }_{i-1j,q}^2+L_{FSO\_i-1}{\sigma }_{i-1j,angle}^2\hfill & \mathrm{if}i=N+1\hfill \end{array}\right.\hfill \\ \hfill {\alpha }_i& ={\left[exp\left({\displaystyle \frac{0.49{\sigma }_{\mathrm{FSO}\_i}^2}{{(1+1.11{\sigma }_{\mathrm{FSO}\_i}^{12/5})}^{7/6}}}\right)-1\right]}^{-1}\hfill \\ \hfill {\beta }_i& ={\left[exp\left({\displaystyle \frac{0.51{\sigma }_{\mathrm{FSO}\_i}^2}{{(1+0.69{\sigma }_{\mathrm{FSO}\_i}^{12/5})}^{5/6}}}\right)-1\right]}^{-1}.\hfill \end{array}$$

(3)

where $jϵ\left\{\mathrm{SU},\mathrm{UU},\mathrm{UD}\right\}$ and $L_{FSO\_i-1}$ is the (i − 1)-th link length, ${\sigma }_{FSO\_i}^2=1.23C_n^2{k}^{7/6}{L}_{FSO\_i}^{11/6}$ is the Rytov variance, $C_n^2$ is the atmospheric refractive index structure parameter, $k={\displaystyle \frac{2\pi }{{\lambda }_F}}$ is the wave number, and ${\lambda }_F$ is the wavelength.

The PDF describing the likelihood of link disruption due to AoA fluctuations has been detailed in reference [15], so the PDF of overall channel gain (denoted as $h_{FSO\_i}$) can be approximated as Equation (4):

$$\begin{array}{cc}\hfill & f_{FSO\_i}\left(h_{FSO\_i}\right)\simeq exp\left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\delta \left(h_{FSO\_i}\right)+\left[1-exp\left({\displaystyle \frac{-{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\right]\hfill \\ & \times {\displaystyle \frac{{\alpha }_i{\beta }_i{g}_{ij}^2}{A_{ij}{h}_{l,FSO\_i}\Gamma \left({\alpha }_i\right)\Gamma \left({\beta }_i\right)}}\times {\mathrm{G}}_{1,3}^{3,0}\left({\displaystyle \frac{{\alpha }_i{\beta }_i}{A_i{h}_{\mathrm{l}\_\mathrm{i}}}}{h}_{FSO\_i}\left|\begin{array}{c}{g}_{ij}^2\\ g_{ij}^2-1,{\alpha }_i-1,{\beta }_i-1\end{array}\right.\right),\hfill \end{array}$$

(4)

the variable $m_i$, which varies according to the type of link, is characterized in detail within reference [16]:

$$m_i=\left\{\begin{array}{cc}1,\hfill & i=1,N+1\\ 2,\hfill & i=2,\dots ,N.\end{array}\right.$$

(5)

In the scenario involving W optical links, each link passes through W reflective elements on the IRS-UAV to reach the next IRS-UAV relay node. This configuration results in an overall instantaneous SNR, as follows:

$$\begin{array}{c}\hfill {\gamma }_{SUMFSO\_i}=\sum _{n=1}^W{\gamma }_{FSO\_i}={\overline{\gamma }}_{FSO\_i}\sum _{n=1}^W{h}_{FSO\_i}^2,\end{array}$$

(6)

${\overline{\gamma }}_{FSO\_i}$ represents the average SNR and $h_{FSO\_i}$ signifies the total channel gain of the initial FSO link. This study adopts an indirect method due to the mathematical complexities involved in directly deriving a closed-form expression for the overall SNR. Specifically, the PDF of $h_{FSO\_i}^2$ is calculated first, followed by leveraging this PDF to ascertain the PDF of $f_{{\gamma }_{SUMFSO\_i}}\left({\gamma }_{SUMFSO\_i}\right)$. The PDF of $h_{FSO\_i}^2$ is derived by setting $K=h_{FSO\_i}^2$:

$$\begin{array}{cc}\hfill & f_K\left(x\right)={\displaystyle \frac{1}{2\sqrt{x}}}exp\left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\delta \left(x^{{\displaystyle \frac{1}{2}}}\right)+\left[1-exp\left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\right]\times {\displaystyle \frac{g_{ij}^2}{2x\Gamma \left({\alpha }_i\right)\Gamma \left({\beta }_i\right)}}\hfill \\ \hfill & \times \left.{\mathrm{G}}_{1,3}^{3,0}\left({\displaystyle \frac{{\alpha }_i{\beta }_i}{A_{ij}{h}_{l,FSO\_i}}}{x}^{{\displaystyle \frac{1}{2}}}\left|\begin{array}{c}{g}_{ij}^2+1\\ g_{ij}^2,{\alpha }_i,{\beta }_i\end{array}\right.\right.\right).\hfill \end{array}$$

(7)

To simplify the computation of the complex SNR expression, this study employs the central limit theorem (CLT) [28], positing that the SNR distribution can be approximated by a Gaussian distribution. Based on this approximation, the expectation (mean), ${\mu }_{FSO}$, and variance, ${\Omega }_{FSO}^2$, of $K=h_{FSO\_i}^2$ are initially calculated. The specific outcomes of these calculations are presented as follows:

$$\begin{array}{cc}\hfill & {\mu }_{FSO}=E\left[K\right]={\displaystyle \frac{{\left(h_{l\_i}{A}_i\right)}^2{g}_{ij}^2\Gamma \left(2+{\alpha }_i\right)\Gamma \left(2+{\beta }_i\right)}{(2+g_{ij}^2)\Gamma \left({\alpha }_i\right)\Gamma \left({\beta }_i\right){\left({\alpha }_i{\beta }_i\right)}^2}}\times \left[1-\exp \left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\right]\hfill \\ \hfill & +\exp \left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}& {\Omega }_{FSO}^2=E\left[K^2\right]-{\left(E\left[K\right]\right)}^2={\displaystyle \frac{{\left(h_{l\_i}{A}_i\right)}^4{g}_{ij}^2\Gamma (4+{\alpha }_i)\Gamma (4+{\beta }_i)}{(4+g_{ij}^2)\Gamma \left({\alpha }_i\right)\Gamma \left({\beta }_i\right){\left({\alpha }_i{\beta }_i\right)}^4}}\times \left[1-\exp \left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\right]\hfill \\ \hfill +& \exp \left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)-{\mu }_{FSO}^2.\hfill \end{array}$$

(9)

The CLT can be effectively applied to accurately estimate the mean and variance of signals reflected by a reflective element in an IRS-UAV system.

$${\mu }_{SUMFSO\_i}={\overline{\gamma }}_{FSO\_i}\times {\mu }_{FSO}\times W,$$

(10)

$${\Omega }_{SUMFSO\_i}^2={\overline{\gamma }}_{FSO\_i}^2\times {\Omega }_{FSO}^2\times W.$$

(11)

Consequently, the PDF of $f_{{\gamma }_{SUMFSO\_i}}\left({\gamma }_{SUMFSO\_i}\right)$ can be derived as follows:

$$\begin{array}{cc}\hfill & f_{{\gamma }_{SUMFSO\_i}}\left({\gamma }_{SUMFSO\_i}\right)\approx {\displaystyle \frac{1}{\sqrt{2\pi {\Omega }_{SUMFSO\_i}^2}}}{e}^{-\mathrm{u}},\hfill \\ & u={\displaystyle \frac{{\left({\gamma }_{SUMFSO\_i}-{\mu }_{SUMFSO\_i})\right.}^2}{2{\Omega }_{SUMFSO\_i}^2}}.\hfill \end{array}$$

(12)

Following the methodology outlined in reference [29], Equation (12) undergoes further mathematical manipulation and integration, resulting in the formulation of Equation (13) as follows:

$$\begin{array}{cc}\hfill & F{\gamma }_{SUMFSO\_i}\left({\gamma }_{SUMFSO\_i}\right)\approx 1-P1-P2,\hfill \\ \hfill & P1={\displaystyle \frac{1}{2\sqrt{\pi }}}\left.{\mathrm{G}}_{1,2}^{2,0}\left({\left({\displaystyle \frac{{\gamma }_{SUMFSO\_i}-{\mu }_{SUMFSO\_i}}{\sqrt{2}{\Omega }_{SUMFSO\_i}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right.\right),\hfill \\ \hfill & P2={\displaystyle \frac{1}{2}}erf\left(-{\displaystyle \frac{{\mu }_{SUMFSO\_i}}{\sqrt{2}{\Omega }_{SUMFSO\_i}}}\right).\hfill \end{array}$$

(13)

2.2. THz Link

To improve the quality of the received signal in the THz link, the transmitting base station S is outfitted with W antennas. The signals emitted by these antennas are relayed to the subsequent IRS-UAV relay node, which is equipped with W reflective elements, thereby enhancing the signal quality in the THz link. On the receiving end, the user terminal D incorporates W receiving antennas and employs MRC as its reception strategy. The system comprises $N+1$ links, with the transmitting base station represented as the 0-th node, and the receiving user terminal as the $(N+1)$-th node. Consequently, the THz signal received at the i-th node of the i-th link can be expressed as

$$\begin{array}{cc}\hfill & y_{THz\_i}\left(t\right)=\left(P_{THz\_i}\sum _{i=1}^w{h}_{THz\_i}\right)S_i+N_{THz\_i},i=1,2,\dots N+1.\hfill \end{array}$$

(14)

where $P_{THz\_i}$ denotes the transmit power of the transmitting base station, and $h_{THz\_i}$ represents the fading coefficient of the terahertz link at the i-th receiving antenna, with $h_{THz\_i}=h_{l,THz\_i}{h}_{a,THz\_i}{h}_{p,THz\_i}{h}_{aoa,THz\_i}$. This channel coefficient is influenced by the cumulative effects of path loss $h_{l,THz\_i}$, channel fading $h_{a,THz\_i}$, antenna misalignment $h_{p,THz\_i}$, and link disruption due to fluctuations in the angle of arrival $h_{aoa,THz\_i}$. The signal-independent additive Gaussian white noise, denoted as $N_{THz\_i}$, has a mean of 0 and a variance of ${\sigma }_0^2$, with $N_{TH{z}_{-}i}\sim \mathcal{N}(0,{\sigma }_0^2)$. Consequently, the SNR at the i-th node is given by ${\gamma }_{\mathrm{THz}\_\mathrm{i}}={\left(P_{\mathrm{THz}\_\mathrm{i}}{\displaystyle \sum _{i=1}^W}{h}_{\mathrm{THz}\_\mathrm{i}}\right)}^2/{\sigma }_0^2$. Subsequently, we analyze the PDF of the channel coefficients for the THz link, considering the impact of each aforementioned factor.

The PDF of link disruption caused by AoA fluctuations was detailed in reference [15]. Based on this, the PDF of the overall channel gain can be derived using the approximate expression provided:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & f_{h_{THz\_i}}\left(h_{THz\_i}\right)\simeq exp\left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\delta \left(h_{THz\_i}\right)+\left[1-\exp \left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\right]\hfill \\ & \times {\displaystyle \frac{g_{ij}^2}{h_{THz\_i}\Gamma \left({\mu }_{THz\_i}\right)}}\times {\mathrm{G}}_{1,2}^{2,0}\left({\displaystyle \frac{{\mu }_{THz\_i}}{{\widehat{h}}_f^{{\alpha }_{THz\_i}}}}{\left({\displaystyle \frac{h_{THz\_i}}{A_{ij}{\mathrm{h}}_{\mathrm{l},THz\_i}}}\right)}^{{\alpha }_{THz\_i}}\left|\begin{array}{c}1;1+{\displaystyle \frac{g_{ij}^2}{{\alpha }_{THz\_i}}}\\ 0,{\displaystyle \frac{{\alpha }_{THz\_i}{\mu }_{THz\_i}}{{\alpha }_{THz\_i}}};0\end{array}\right.\right).\hfill \end{array}\end{array}$$

(15)

According to the CLT, after the signal traverses through W antennas and is reflected by W IRS elements at the IRS-UAV relay node, the PDF and CDF of the SNR can be derived through a methodology analogous to that previously employed for determining the SNR of the FSO link, as detailed below:

$$\begin{array}{ccc}\hfill & f_{{\gamma }_{SUMTHz\_i}}\left({\gamma }_{SUMTHz\_i}\right)\hfill & \hfill \approx {\displaystyle \frac{1}{\sqrt{2\pi {\Omega }_{SUMTHz\_i}^2}}}{e}^{-{\displaystyle \frac{{\left({\gamma }_{SUMTHz\_i}-{\mu }_{SUMTHz\_i})\right.}^2}{2{\Omega }_{SUMTHz\_i}^2}}},\end{array}$$

(16)

$$\begin{array}{cc}\hfill & F_{{\gamma }_{SUMTHz\_i}}\left({\gamma }_{SUMTHz\_i}\right)\approx 1-{\displaystyle \frac{1}{2\sqrt{\pi }}}{\mathrm{G}}_{1,2}^{2,0}\left.\left({\left({\displaystyle \frac{{\gamma }_{SUMTHz\_i}-{\mu }_{SUMTHz\_i}}{\sqrt{2{\Omega }_{SUMTHz\_i}^2}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right.\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}erf\left(-{\displaystyle \frac{{\mu }_{SUMTHz\_i}}{\sqrt{2}{\Omega }_{SUMTHz\_i}}}\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & {\mu }_{THz}=E\left[Z\right]={\displaystyle \frac{g_{ij}^2\Gamma \left(\frac{2}{{\alpha }_{THz\_i}}+\frac{{\alpha }_{THz\_i}{\mu }_{THz\_i}}{{\alpha }_{THz\_i}}\right)}{\Gamma \left({\mu }_{THz\_i}\right)\left(2+g_{ij}^2\right)}}{\left({\displaystyle \frac{{\widehat{h}}_f{A}_{ij}{h}_{l,THz\_i}}{{{\mu }_{THz\_i}}^{\frac{1}{{\alpha }_{THz\_i}}}}}\right)}^2\times \left[1-exp\left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\right]\hfill \\ \hfill & +exp(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & {\Omega }_{THz}^2=E\left[Z^2\right]-{\left(E\left[Z\right]\right)}^2={\displaystyle \frac{g_{ij}^2\Gamma \left(\frac{4}{{\alpha }_{THz\_i}}+\frac{{\alpha }_{THz\_i}{\mu }_{THz\_i}}{{\alpha }_{THz\_i}}\right)}{\Gamma \left({\mu }_{THz\_i}\right)\left(4+g_{ij}^2\right)}}{\left({\displaystyle \frac{{\widehat{h}}_f{A}_{ij}{\mathrm{h}}_{1,THz\_i}}{{\mu }_{THz\_i}^{\frac{1}{{\alpha }_{THz\_i}}}}}\right)}^4\hfill \\ \hfill & \times \left[1-exp\left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)\right]+exp\left(-{\displaystyle \frac{{\psi }_{AoA,i}^2}{2m_i{\sigma }_{i-1j,angle}^2}}\right)-{{\mu }_{THz}}^2,\hfill \end{array}$$

(19)

$${\mu }_{SUMTHz\_i}={\overline{\gamma }}_{THz\_i}\times {\mu }_{THz}\times W,$$

(20)

$${\Omega }_{SUMTHz\_i}^2={\overline{\gamma }}_{THz\_i}^2\times {\Omega }_{THz}^2\times W.$$

(21)

3. Performance Analysis

In this section, we derive closed-form expressions for the outage probability, average BER, and average channel capacity of the proposed UAV-IRS hybrid multi-hop FSO/THz link system to analyze its performance.

3.1. Outage Probability

Outage probability, a key performance metric in wireless communications, quantifies the likelihood of data transmission interruption when the end-to-end SNR $\gamma$ falls below a specified threshold ${\gamma }_{th}$ [30,31]. Thus, the outage probability of the i-th FSO link of the i-th node can be obtained as

$$P_{out}^{FSO\_i}=Pr\left(\gamma \le {\gamma }_{th}^{FSO}\right)=F_{{\gamma }_{SUMFSO\_i}}\left({\gamma }_{th}^{FSO}\right).$$

(22)

Furthermore, the end-to-end outage probability for FSO communication can be calculated as follows:

$$P_{out}^{FSO}=1-\prod _{i=1}^N(1-P_{out}^{FSO\_i}).$$

(23)

Similarly, the outage probability for the i-th THz link, which connects the i-th node and the i-th node, can be calculated as follows:

$$P_{out}^{THz\_i}=Pr\left(\gamma \le {\gamma }_{th}^{THz}\right)=F_{{\gamma }_{SUMTHz\_i}}\left({\gamma }_{th}^{THz}\right).$$

(24)

The end-to-end outage probability for the THz system can be calculated as follows:

$$P_{out}^{THz}=1-\prod _{i=1}^N(1-P_{out}^{THz\_i}).$$

(25)

Subsequently, we analyze the outage probability considering two distinct switching strategies:

• Initially, we conduct a performance analysis focusing on the system outage probability under the hard-switching strategy. In this setup, the THz link is activated when the condition ${\gamma }_{THz\_i}\ge {\gamma }_{th,i}$ is met. In contrast, when conditions ${\gamma }_{THz\_i}<{\gamma }_{th,i}$ and ${\gamma }_{FSO\_i}\ge {\gamma }_{th,i}$ are met, a feedback mechanism activates the FSO link and deactivates the THz link. Consequently, the end-to-end outage probability of the hybrid THz/FSO IRS-UAV system, utilizing this specific switching strategy, can be expressed as [32]

$$\begin{array}{ccc}\hfill & P_{out}^1=1-\prod _{i=1}^N\left(1-P_{out}^{FSO\_i}\times P_{out}^{THz\_i}\right)\hfill & \hfill =1-\prod _{i=1}^N\left(1-F_{{\gamma }_{SUMFSO\_i}}\left({\gamma }_{th}^{FSO}\right)\times F_{{\gamma }_{SUMTHz\_i}}\left({\gamma }_{th}^{THz}\right)\right).\end{array}$$

(26)

• In this section, we assume that a soft-switch solution is used. Furthermore, the end-to-end outage probability using selective combination (SC) technology is expressed as

$$P_{out}^2=P_{out}^{P1\_i}\left({\gamma }_{th,l}^{FSO},{\gamma }_{th,u}^{FSO}\right)\times P_{out}^{THz\_i}\left({\gamma }_{th}^{THz}\right),$$

(27)

where

$$P_{out}^{P1\_i}({\gamma }_{th,l}^{FSO},{\gamma }_{th,u}^{FSO})=P_{Low}^{FSO\_i}+P_{Med}^{FSO\_i}·{\displaystyle \frac{P_{Low}^{FSO\_i}}{P_{Hi}^{FSO\_i}+P_{Low}^{FSO\_i}}},$$

(28)

$$P_{Low}^{FSO\_i}=P_{out}^{FSO\_i}\left({\gamma }_{th,l}^{FSO}\right),P_{Hi}^{FSO\_i}=1-P_{out}^{FSO\_i}\left({\gamma }_{th,u}^{FSO}\right),$$

(29)

$$P_{Med}^{FSO\_i}=P_{out}^{FSO\_i}\left({\gamma }_{th,u}^{FSO}\right)-P_{out}^{FSO\_i}\left({\gamma }_{th,l}^{FSO}\right).$$

(30)

3.2. Average Bit Error Rate

The average BER serves as a direct indicator of the modulation method’s influence on the performance of the communication system. The equation for determining the BER is provided in reference [33] as follows:

$$P_e={\displaystyle \frac{q^p}{2\Gamma \left(p\right)}}{\int }_0^{\infty }{\gamma }^{p-1}{e}^{-q\gamma }{F}_{\gamma }\left(\gamma \right)d\gamma .$$

(31)

In this study, according to reference [34], BPSK modulation is employed, setting the parameters as $p=0.5$ and $q=1$. To calculate the BER for the FSO link, we first process the CDF of the FSO link; according to Equation (13), it can be obtained that the expression ${\displaystyle \frac{1}{2}}\mathrm{erf}\left(-{\displaystyle \frac{{\mu }_{\mathrm{SUMFSO}\_i}}{\sqrt{2}{\Omega }_{\mathrm{SUMFSO}\_i}}}\right)$ is considered a constant. To simplify the computation process, this entire expression is denoted by T. Consequently, the CDF of the FSO link is reformulated into a novel expression:

$$\begin{array}{ccc}\hfill & F_{{\gamma }_{\mathrm{SUM}FSO\_i}}\left({\gamma }_{\mathrm{SUM}FSO\_i}\right)=1-{\displaystyle \frac{1}{2\sqrt{\pi }}}\hfill & \hfill \times {\mathrm{G}}_{1,2}^{2,0}\left.\left({\left({\displaystyle \frac{{\gamma }_{\mathrm{SUMFSO}\_i}-{\mu }_{\mathrm{SUMFSO}\_i}}{\sqrt{2}{\Omega }_{SUMFSO\_i}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right.\right)+T.\end{array}$$

(32)

The end-to-end PDF for the FSO link is as follows:

$$\begin{array}{cc}\hfill & F_{{\gamma }_{SUMFSO}}\left({\gamma }_{SUMFSO}\right)=1-{\left(1-F_{{\gamma }_{SUMFSO\_i}}\left({\gamma }_{SUMFSO\_i}\right)\right)}^N\hfill \\ \hfill & =1-\sum _{k=0}^N\left(\begin{array}{c}N\\ k\end{array}\right){\left(-T\right)}^{N-k}\times {\displaystyle \frac{1}{2\sqrt{\pi }}}{\left({\mathrm{G}}_{1,2}^{2,0}\left({\left({\displaystyle \frac{{\gamma }_{\mathrm{SUMFSO}\_i}-{\mu }_{\mathrm{SUMFSO}\_i}}{\sqrt{2}{\Omega }_{SUMFSO\_i}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right)\right)}^k.\hfill \end{array}$$

(33)

So the average symbol BER of the FSO link is

$$\begin{array}{ccc}\hfill & P_{efso}={\displaystyle \frac{q^p}{2\Gamma \left(p\right)}}{\int }_0^{\infty }{{\gamma }_{SUMFSO\_i}}^{p-1}{e}^{-q{\gamma }_{SUMFSO\_i}}\hfill & \hfill \times F_{{\gamma }_{SUMFSO}}\left({\gamma }_{SUMFSO}\right)d{\gamma }_{SUMFSO\_i}.\end{array}$$

(34)

It can further be written as $P_{efso}=I_1-I_2$, where $I_1$ and $I_2$ are denoted as follows:

$$I_1={\displaystyle \frac{q^p}{2\Gamma \left(p\right)}}{\int }_0^{\infty }{\gamma }^{p-1}{e}^{-q\gamma }d\gamma ,$$

(35)

$$\begin{array}{cc}\hfill & I_2={\displaystyle \frac{q^p}{2\Gamma \left(p\right)}}{\int }_0^{\infty }{{\gamma }_{SUMFSO\_i}}^{p-1}{e}^{-q{\gamma }_{SUMFSO\_i}}\sum _{k=0}^N\left(\begin{array}{c}\hfill N\\ \hfill k\end{array}\right)\times {\left(-T\right)}^{N-k}\hfill \\ & {\left({\displaystyle \frac{1}{2\sqrt{\pi }}}{\mathrm{G}}_{1,2}^{2,0}\left({\left({\displaystyle \frac{{\gamma }_{\mathrm{SUMFSO}\_i}-{\mu }_{\mathrm{SUMFSO}\_i}}{\sqrt{2}{\Omega }_{SUMFSO\_i}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right)\right)}^{k}d{\gamma }_{SUMFSO\_i}.\hfill \end{array}$$

(36)

Drawing on the methodology outlined in reference [35], the calculation of the $I_1$ integral can be performed as follows:

$$\begin{array}{c}\hfill I_1={\displaystyle \frac{1}{q^p}}\Gamma \left(p\right).\end{array}$$

(37)

Computing the integral of $I_2$ is inherently complex, making the derivation of its closed-form expression particularly challenging when relying on conventional integration methods. To address this issue, the Gauss–Laguerre integration criterion is employed herein to approximate the expression for the average BER in an end-to-end closed scenario:

$$\begin{array}{cc}\hfill & P_{efso}\approx {\displaystyle \frac{1}{q^p}}\Gamma \left(p\right)-{\displaystyle \frac{q^p}{2\Gamma \left(p\right)}}\sum _{k=0}^N\left(\begin{array}{c}N\\ k\end{array}\right){\left(-T\right)}^{N-k}{\left({\displaystyle \frac{1}{2}}\right)}^k\times {\left({\displaystyle \frac{1}{q}}\right)}^p\hfill \\ & \times \sum _{i=1}^n{\phi }_{FSO\_i}\times {\left({\mathrm{G}}_{1,2}^{2,0}\left({\left({\displaystyle \frac{{\gamma }_{\mathrm{SUMFSO}\_i}-{\mu }_{\mathrm{SUMFSO}\_i}}{\sqrt{2}{\Omega }_{SUMFSO\_i}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right)\right)}^k,\hfill \end{array}$$

(38)

$$\begin{array}{c}\hfill {\phi }_{FS{O}_{-}i}={\displaystyle \frac{\Gamma (n+p){\gamma }_{SUMFSO\_i}}{n!{\left(n+1\right)}^2{\left[L_{n+1}^{(p-1)}\left({\gamma }_{SUMFSO\_i}\right)\right]}^2}}.\end{array}$$

(39)

The same reasoning is available:

$$\begin{array}{cc}\hfill & P_{eTHz}\simeq {\displaystyle \frac{1}{q^p}}\Gamma \left(p\right)-{\displaystyle \frac{q^p}{2\Gamma \left(p\right)}}\sum _{k=0}^N\left(\begin{array}{c}N\\ k\end{array}\right){\left(-T\right)}^{N-k}{\left({\displaystyle \frac{1}{2}}\right)}^k\times {\left({\displaystyle \frac{1}{q}}\right)}^p\hfill \\ & \times \sum _{i=1}^n{\phi }_{THz\_i}\times \left({\mathrm{G}}_{1,2}^{2,0}\left({\left({\displaystyle \frac{{\gamma }_{SUMTHz\_i}-{\mu }_{SUMTHz\_i}}{\sqrt{2}{\Omega }_{SUMTHz\_i}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right)\right),\hfill \end{array}$$

(40)

$${\phi }_{TH{z}_{-}i}={\displaystyle \frac{\Gamma (n+p){\gamma }_{SUMTHz\_i}}{n!{\left(n+1\right)}^2{\left[L_{n+1}^{(p-1)}\left({\gamma }_{SUMTHz\_i}\right)\right]}^2}}.$$

(41)

The expressions for the average symbol BER corresponding to hard-switching and soft-switching strategies are presented below:

$$\begin{array}{c}\hfill P_e^1={\displaystyle \frac{J1}{J2}},\end{array}$$

(42)

$$P_e^2={\displaystyle \frac{J3}{J4}}.$$

(43)

The expressions for $J1$ and $J2$ are as follows:

$$\begin{array}{cc}\hfill & J1=\left(\prod _{i=1}^N\left(1-P_{out}^{FSO\_i}\right)\right)P_{efso}+\prod _{i=1}^N{P}_{out}^{FSO\_i}\left({\gamma }_{th}^{FSO}\right)\times \left(\prod _{i=1}^N\left(1-P_{out}^{THz\_i}\right)\right)P_{eTHz},\hfill \end{array}$$

(44)

$$J2=\prod _{i=1}^N\left(1-P_{out}^{FSO}\times P_{out}^{THz}\right).$$

(45)

The expressions for $J3$ and $J4$ are as follows:

$$\begin{array}{cc}\hfill & J3=\left(\prod _{i=1}^N\left(1-P_{out}^{P1\_i}\left({\gamma }_{th,l}^{FSO},{\gamma }_{th,u}^{FSO}\right)\right)\right)P_{efso}+\prod _{i=1}^N{P}_{out}^{P1\_i}\left({\gamma }_{th,l}^{FSO},{\gamma }_{th,u}^{FSO}\right)\left(\prod _{i=1}^N\left(1-P_{out}^{THz\_i}\right)\right)P_{eTHz},\hfill \end{array}$$

(46)

$$J4=\prod _{i=1}^N(1-P_{out}^{2\_i}).$$

(47)

3.3. Average Channel Capacity

According to the findings in references [36,37], the formula for average channel capacity is represented by the variable $\overline{C}=E\left[{log}_2(1+\gamma )\right]$, where $E\left(·\right)$ serves as the mathematical expectation operator. Consequently, this allows for the definition of the average channel capacity:

$$\begin{array}{cc}\hfill & C={\int }_0^{\infty }{log}_2(1+\gamma )f_{\gamma }\left(\gamma \right)d\gamma ={\displaystyle \frac{1}{ln2}}{\int }_0^{\infty }{\displaystyle \frac{F_{\gamma }^C\left(\gamma \right)}{1+\gamma }}d\gamma ,\hfill \end{array}$$

(48)

where $F_{\gamma }^C\left(\gamma \right)=1-F_{\gamma }\left(\gamma \right)$. The channel capacity of the FSO link is derived from Equation (48) as presented above:

$$\begin{array}{cc}\hfill & C_{FSO}\left({\gamma }_{th}^{FSO}\right)={\displaystyle \frac{1}{ln2}}{\int }_{{\gamma }_{th}^{FSO}}^{\infty }{log}_2(1+\gamma )f_{\gamma }\left(\gamma \right)d\gamma ={\displaystyle \frac{1}{ln2}}{\int }_{{\gamma }_{th}^{FSO}}^{\infty }{\displaystyle \frac{1}{1+{\gamma }_{SUMFSO\_i}}}\hfill \\ \hfill & \times \left({\left(1-F_{{\gamma }_{SUMFSO\_i}}\left({\gamma }_{SUMFSO\_i}\right)\right)}^N\right)d{\gamma }_{SUMFSO\_i}.\hfill \end{array}$$

(49)

Transforming the aforementioned equation in its entirety to satisfy the condition ${\gamma }_{SUMFSO\_i}-{\gamma }_{th}^{FSO}=t$ yields the equation

$$\begin{array}{cc}\hfill & C_{FSO}\left({\gamma }_{th}^{FSO}\right)={\displaystyle \frac{1}{ln2}}\times \sum _{k=0}^N\left(\begin{array}{c}N\\ k\end{array}\right){\left(-T\right)}^{N-k}\times {\int }_0^{\infty }{\displaystyle \frac{1}{1+t+{\gamma }_{th}^{FSO}}}\hfill \\ \hfill & {\left({\displaystyle \frac{1}{2\sqrt{\pi }}}{\mathrm{G}}_{1,2}^{2,0}\left({\left({\displaystyle \frac{t+{\gamma }_{th}^{THz}-{\mu }_{SUMTH{z}_{-}i}}{\sqrt{2}{\Omega }_{SUMTH{z}_{-}i}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right)\right)}^{k}d{\gamma }_{SUMFSO\_i}.\hfill \end{array}$$

(50)

Modifying the preceding equation results in

$$\begin{array}{cc}\hfill & C_{FSO}\left({\gamma }_{th}^{FSO}\right)={\displaystyle \frac{1}{ln2}}\sum _{k=0}^N\left(\begin{array}{c}N\\ k\end{array}\right)\times {\left(-T\right)}^{N-k}\times {\int }_0^{\infty }{e}^{-t}\times e^t\times {\displaystyle \frac{1}{t+{\gamma }_{th}^{FSO}+1}}\hfill \\ \hfill & \times {\left({\displaystyle \frac{1}{2\sqrt{\pi }}}{\mathrm{G}}_{1,2}^{2,0}\left({\left({\displaystyle \frac{t+{\gamma }_{th}^{THz}-{\mu }_{SUMTH{z}_{-}i}}{\sqrt{2}{\Omega }_{SUMTH{z}_{-}i}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right)\right)}^{k}dt.\hfill \end{array}$$

(51)

The following steps are taken to process the above equation, as follows:

$$\begin{array}{cc}\hfill & g_{\mathrm{FSO}}\left(t\right)=e^t\times {\displaystyle \frac{1}{t+{\gamma }_{th}^{FSO}+1}}\times {\left({\displaystyle \frac{1}{2\sqrt{\pi }}}{\mathrm{G}}_{1,2}^{2,0}\left({\left({\displaystyle \frac{t+{\gamma }_{th}^{FSO}-{\mu }_{SUMFSO\_i}}{\sqrt{2}{\Omega }_{SUMFSO\_i}^2}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right)\right)}^k.\hfill \end{array}$$

(52)

Its approximate expression is derived employing the Gaussian–Laguerre product formula:

$$\begin{array}{c}\hfill C_{FSO}={\displaystyle \frac{1}{ln2}}\sum _{k=0}^N\left(\begin{array}{c}N\\ k\end{array}\right){\left(-T\right)}^{N-k}{\int }_0^{\infty }{e}^{-t}{g}_{FSO}\left(t\right)dt\approx \sum _{k=0}^N\left(\begin{array}{c}N\\ k\end{array}\right){\left(-T\right)}^{N-k}\sum _{i=1}^n{w}_{CFSO\_i}{g}_{FSO}\left(t_i\right),\end{array}$$

(53)

$$w_{CFSO\_i}={\displaystyle \frac{\Gamma \left(n\right)t_i}{n!{(n+1)}^2{\left[L_{n+1}^{(-1)}\left(t_i\right)\right]}^2}}.$$

(54)

Similarly, the expression for the channel capacity closure of THz link multi-hop configuration can be derived as follows:

$$\begin{array}{c}\hfill C_{\mathrm{THz}}={\displaystyle \frac{1}{ln2}}\sum _{k=0}^N\left(\begin{array}{c}N\\ k\end{array}\right){\left(-T\right)}^{N-k}{\int }_0^{\infty }{e}^{-t}{g}_{THz}\left(t\right)dt=\sum _{k=0}^N\left(\begin{array}{c}N\\ k\end{array}\right){\left(-T\right)}^{N-k}\sum _{i=1}^n{w}_{CTHz\_i}{g}_{THz}\left(t_i\right),\end{array}$$

(55)

$$w_{CTHz\_i}={\displaystyle \frac{\Gamma \left(n\right)t_i}{n!{(n+1)}^2{\left[L_{n+1}^{(-1)}\left(t_i\right)\right]}^2}},$$

(56)

$$\begin{array}{cc}\hfill & g_{THE}\left(t_i\right)={\displaystyle \frac{1}{t_i+{\gamma }_{ih}^{THz}+1}}{e}^{t_i}\times {\left.\left.\left({\displaystyle \frac{1}{2}}{\mathrm{G}}_{1,2}^{2,0}\left({\left({\displaystyle \frac{t+{\gamma }_{th}^{THz}-{\mu }_{SUMTH{z}_{-}i}}{\sqrt{2}{\Omega }_{SUMTH{z}_{-}i}}}\right)}^2\left|\begin{array}{c}1\\ 0,{\displaystyle \frac{1}{2}}\end{array}\right.\right.\right.\right)\right)}^k.\hfill \end{array}$$

(57)

According to the findings presented in reference [38], the channel capacities for hard switching and soft switching are determined as follows:

$$C_1=C_{FSO}\left({\gamma }_{th}^{FSO}\right)+P^{FSO}\left({\gamma }_{th}^{THz}\right)·C_{THz}\left({\gamma }_{th}^{FSO}\right),$$

(58)

$$w_{CTHz\_i}={\displaystyle \frac{\Gamma \left(n\right)t_i}{n!{(n+1)}^2{\left[L_{n+1}^{(-1)}\left(t_i\right)\right]}^2}},$$

(59)

$$\begin{array}{cc}\hfill & C_2=C_{FSO}\left({\gamma }_{th,u}^{FSO}\right)+P_{out}^{P1}({\gamma }_{th,l}^{FSO},{\gamma }_{th,u}^{FSO})\times C_{THz}\left({\gamma }_{th}^{THz}\right)+[C_{FSO}\left({\gamma }_{th,l}^{FSO}\right)-C_{FSO}\left({\gamma }_{th,u}^{FSO}\right)]\times {\displaystyle \frac{P_{Hi}^{P1}}{P_{Hi}^{P1}+P_{Low}^{P1}}},\hfill \end{array}$$

(60)

$$P_{out}^{P1}({\gamma }_{th,l}^{FSO},{\gamma }_{th,u}^{FSO})=P_{Low}^{P1}+P_{Med}^{P1}\times {\displaystyle \frac{P_{Low}^{P1}}{P_{Hi}^{P1}+P_{Low}^{P1}}}.$$

(61)

4. Numerical Results

In this section, we assess the performance of the UAV-IRS hybrid multi-hop FSO/THz link system utilizing the mathematical expressions formulated in Section 3. To streamline the analysis, as both FSO and THz links are relayed to subsequent nodes via identical IRS-UAV repeaters, the link pointing error parameters for FSO and THz remain consistent throughout the analysis.

Table 2. Parameter settings for FSO, THz, and RF links without special instructions.

	Parameter	Value
FSO Link	Angle of arrival	${\psi }_{AOA}=4\mathrm{mrad}$
	Wavelength	${\lambda }_F=1550\mathrm{nm}$
	Link length	$L_{\mathrm{FSO}\_i}=250\mathrm{m}$
	Visibility	$V=20\mathrm{km}$
	Moderate turbulence	$C_n^2=1\times {10}^{-13}{\mathrm{m}}^{-2/3}$
	Strong turbulence	$C_n^2=1\times {10}^{-12}{\mathrm{m}}^{-2/3}$
	Small pointing errors	${\sigma }_{(i-1)j,u}=0.25\mathrm{m}$, ${\sigma }_{ij,u}=0.25\mathrm{m}$, ${\sigma }_{(i-1)j,\mathrm{angle}}=0.4\mathrm{m}$
	Large pointing errors	${\sigma }_{(i-1)j,u}=0.5\mathrm{m}$, ${\sigma }_{ij,u}=0.5\mathrm{m}$, ${\sigma }_{(i-1)j,\mathrm{angle}}=0.8\mathrm{m}$
THz Link	Frequency	$f_T=100\mathrm{GHz}$
	Link length	$d_T=200\mathrm{m}$
	Antenna gain	$G_t^T=G_r^T=55\mathrm{dBi}$
	Standard atmospheric conditions	$p=\mathrm{101,325}\mathrm{Pa}$, $T=25{ }^{\circ }\mathrm{C}$, $\varphi =50\%$
	Moderate turbulence	${\alpha }_T=1.7$, ${\mu }_T=1.7$
	Strong turbulence	${\alpha }_T=1.7$, ${\mu }_T=1.5$
	Small pointing errors	${\sigma }_{(i-1)j,u}=0.25\mathrm{m}$, ${\sigma }_{ij,u}=0.25\mathrm{m}$, ${\sigma }_{(i-1)j,\mathrm{angle}}=0.4\mathrm{m}$
	Large pointing errors	${\sigma }_{(i-1)j,u}=0.5\mathrm{m}$, ${\sigma }_{ij,u}=0.5\mathrm{m}$, ${\sigma }_{(i-1)j,\mathrm{angle}}=0.8\mathrm{m}$

shows typical system data used, including performance parameters of FSO and the THz link if not otherwise specified. New conditions are specified if they are introduced during the qualitative analysis. In addition, the numerical simulation plots below, whenever case 1 and case 2 are involved, are for the following conditions: Case 1 is moderately turbulent with small pointing errors; case 2 is strongly turbulent with large pointing errors.

The outage probability of the UAV-based IRS hybrid multi-hop FSO/THz link system is shown in Figure 2. Figure 2 illustrates the case where the number of links N = 4, where ${\gamma }_{th,l}^{FSO}=3\mathrm{dB}$, ${\gamma }_{th,u}^{FSO}=7\mathrm{dB}$, ${\gamma }_{th}^{FSO}={\gamma }_{th}^{THz}=4\mathrm{dB}$, and the number of IRS reflective elements $W=24$. The system’s outage probability fluctuates with the average SNR under various turbulence conditions, pointing errors, and operational modes, as depicted in the figure. It is observed that, holding turbulence conditions and modes of operation constant, the system’s outage probability consistently diminishes as the average SNR increases. This indicates that enhancing the link system’s SNR can significantly improve the system’s communication performance. At the same average SNR, for identical modes (FSO multi-hop link and THz multi-hop link), the outage probability performance closely resembles that observed under conditions of medium turbulence with weak pointing errors, as well as under strong turbulence with strong pointing errors. This observation indicates that IRS can effectively reduce the adverse effects of turbulence and pointing errors on system performance. Moreover, the outage probability when employing hard switching is superior to that observed in both FSO and THz links. Specifically, when the FSO link’s communication quality is subpar and fails to meet the required performance standards, the system seamlessly transitions to the THz link to maintain communication reliability. Furthermore, under identical SNR conditions, the performance of the outage probability with soft switching surpasses that of hard switching. This suggests that soft switching not only extends the operational duration of the FSO link but also minimizes the negative consequences of frequent transitions associated with hard switching on the system’s outage probability performance.

In the simulation depicted in Figure 2, the system employing soft switching demonstrates superior performance regarding outage probability. Consequently, the subsequent phase involves utilizing soft switching to explore additional aspects of the system’s performance. Figure 3 presents a sequence of experiments designed to assess the system’s response to significant turbulence and the impact of pronounced pointing errors, where the parameters are ${\gamma }_{th,l}^{FSO}=3\mathrm{dB}$, ${\gamma }_{th,u}^{FSO}=7\mathrm{dB}$, and ${\gamma }_{th}^{FSO}={\gamma }_{th}^{THz}=4\mathrm{dB}$, with the number of links set to N = 4, 6, 8, and 10. The system’s performance is compared both with and without the implementation of the IRS scheme. The analysis reveals a decline in system performance as the number of links increases. This decline is attributed to the heightened probability of an error occurring in one of the relays when employing a serial IRS-UAV as a relay, consequently elevating the system’s outage probability. Moreover, the system utilizing the IRS scheme outperforms its counterpart without the IRS, notably exhibiting several orders of magnitude lower outage probability when IRS numbers = 4 and the number of links N = 10, compared to a non-IRS setup with N = 4 links. This indicates that IRS deployment can significantly mitigate the adverse effects of increasing link numbers on system performance. Thus, by strategically setting the IRS numbers and link count, it is feasible to extend communication coverage while striving for optimal outage probability performance.

Based on the findings presented in Figure 4, this study examines the performance of the system’s outage probability when employing a soft-switching scheme under various conditions, including different levels of turbulence intensity, pointing errors, and the number of reflecting surface elements. Where the parameters are ${\gamma }_{th,l}^{FSO}=3\mathrm{dB}$, ${\gamma }_{th,u}^{FSO}=7\mathrm{dB}$, and ${\gamma }_{th}^{FSO}={\gamma }_{th}^{THz}=4\mathrm{dB}$, with the number of links set to N = 4. As depicted in Figure 4, under consistent turbulence intensity and pointing error conditions, the system’s outage probability performance notably enhances as the number of IRS elements increases. This improvement highlights the significant role of increasing IRS elements in bolstering the system’s resilience, by facilitating additional transmission pathways and signal amplification, thereby mitigating the adverse impacts of turbulence and pointing inaccuracies. Remarkably, at a constant SNR, the system’s performance under severe turbulence and pronounced pointing errors, with an IRS count of 16, parallels that observed under milder turbulence and pointing conditions with just 4 IRS elements. This parallelism underscores the potential of escalating IRS counts to counteract the detrimental effects of turbulence and pointing errors effectively. Consequently, scaling up the IRS array can markedly elevate system performance, allowing for tailored IRS configurations based on the severity of turbulence and pointing errors encountered in diverse communication contexts. This strategic adjustment is crucial for optimizing system architecture and enhancing overall communication efficacy.

The analysis presented in Figure 5 investigates the scenario where the number of links (N) is eight and the number of reflective surface elements (W) is four, where the parameters are ${\gamma }_{th,l}^{FSO}=3\mathrm{dB}$, ${\gamma }_{th,u}^{FSO}=7\mathrm{dB}$, and ${\gamma }_{th}^{FSO}={\gamma }_{th}^{THz}=4\mathrm{dB}$. The impact of various modes, pointing errors, and turbulence strengths on the BER is analyzed. Figure 5 reveals that the system’s BER performance is superior under conditions of moderate turbulence and minimal pointing errors compared to scenarios involving strong turbulence and significant pointing errors within the same mode. This observation underscores the detrimental effects of turbulence and pointing errors on system efficiency. Nonetheless, with an increase in the SNR, the BER performance under both moderate and strong turbulence and pointing errors converges, particularly when employing FSO multi-hop links and THz multi-hop links. This convergence suggests that the adverse impacts of turbulence and pointing errors can be markedly mitigated through the use of IRS. Furthermore, Figure 5 also highlights the performance disparities across different modes, with the soft-switching scheme yielding the best BER performance, followed by the hard-switching scheme. This indicates that the adoption of soft- and hard-switching schemes can significantly enhance the system’s BER performance, thereby improving the quality of communication transmission.

The findings presented in Figure 6 explore the impact of varying turbulence intensities, pointing errors, and the quantity of reflective surface elements on the system’s BER within a soft-switching scheme context. Here, “IRS-Number” refers to the count of reflective surface elements. The parameters are ${\gamma }_{th,l}^{FSO}=3\mathrm{dB}$, ${\gamma }_{th,u}^{FSO}=7\mathrm{dB}$, and ${\gamma }_{th}^{FSO}={\gamma }_{th}^{THz}=4\mathrm{dB}$, with the number of links set to N = 4. The figure demonstrates that the system’s BER performance progressively enhances with a higher count of IRS elements, maintaining constant turbulence strength and pointing error levels. This improvement suggests that increasing the IRS element count can markedly boost system performance, particularly under conditions of strong turbulence and significant pointing errors. Notably, a configuration with 16 IRS elements outperforms one with 8 IRS elements in scenarios of both moderate turbulence and minimal pointing errors. Consequently, an increase in IRS elements can effectively mitigate the adverse impacts of turbulence and pointing errors on system performance. In environments where turbulence and pointing errors are prevalent, adjusting the number of reflective surface elements upwards can optimize system performance. Elevating the IRS count serves to diminish the detrimental effects of turbulence and pointing errors, thereby enhancing the system’s BER performance.

In Figure 7, under the action of strong turbulence and strong pointing errors, we set the number of links to be N = 4, 6, 8, and 10, and analyzed the effect on the average BER of the system with or without the use of an IRS and with different numbers of links. From the figure, we can observe that as the number of links increases, the performance of the system becomes progressively worse, this is because when serial relaying is used, the increase in the number of relays increases the probability of error in a particular relay, which leads to a decrease in the performance of the system. In addition, we can observe that the performance of the system with the IRS scheme is better than the performance of the system without the IRS scheme; especially, when IRS number = 4 and the number of links N = 10, the BER performance is orders of magnitude lower than that without IRS and the number of links N = 4. This indicates that the use of IRS can largely improve the performance degradation due to the increase in the number of links.

In Figure 8, for the number of links N = 10, where the parameters are ${\gamma }_{th,l}^{FSO}=3\mathrm{dB}$, ${\gamma }_{th,u}^{FSO}=7\mathrm{dB}$, and ${\gamma }_{th}^{FSO}={\gamma }_{th}^{THz}=4\mathrm{dB}$ and the number of reflective surface elements W = 4, this study investigates the impact of varying operational modes, pointing errors, and turbulence strengths on the system’s channel capacity. Analysis of the data reveals that optimal channel capacity is achieved under conditions of moderate turbulence and minimal pointing errors, within the same operational mode. This finding suggests that the system’s channel capacity is significantly higher under conditions of reduced turbulence and pointing errors. Additionally, the performance of channel capacity varies across different modes; specifically, the soft-switching scheme outperforms the hard-switching scheme, indicating that employing soft-switching and hard-switching strategies can enhance the system’s channel capacity. Furthermore, the channel capacity of a standalone THz link is observed to be inferior to that in the other three modes. Because FSOs employ the visible or near-infrared wavelengths, which have higher frequencies than THz bands, and so, theoretically give a broader bandwidth, this discrepancy can be linked to Shannon’s theorem. This leads to a reduced terahertz channel capacity at an equivalent SNR. As SNR increases, the channel capacity of the FSO link converges with those observed during hard and soft switching. This convergence is explained by the system’s increased reliance on the FSO link for communication at higher SNRs, reducing the likelihood of switching to the THz link. Consequently, with rising SNR, the channel capacity of the FSO link approaches that observed during both switching modes.

According to the results in Figure 9, a soft-switching scheme is considered under the influence of different turbulence intensities, pointing errors, and the number of reflective surface elements. Where the parameters are ${\gamma }_{th,l}^{FSO}=3\mathrm{dB}$, ${\gamma }_{th,u}^{FSO}=7\mathrm{dB}$, and ${\gamma }_{th}^{FSO}={\gamma }_{th}^{THz}=4\mathrm{dB}$, with the number of links set to N = 10. The analysis of the figure reveals that, holding the degree of turbulence and pointing errors constant, the system’s channel capacity progressively increases with the addition of IRS. This finding underscores the significant role that increasing the number of IRS elements plays in enhancing the system’s channel capacity. Notably, under conditions of strong turbulence and pronounced pointing errors, the system’s performance with sixteen IRS elements surpasses that under mild turbulence and slight pointing errors with only eight IRS elements. Thus, it is evident that increasing the IRS count not only boosts the system’s channel capacity but also mitigates the adverse effects of turbulence and pointing errors on system performance. Consequently, selecting the optimal number of IRS elements based on the specific requirements of the communication scenario can optimize the IRS’s utilization efficiency.

In the context of strong turbulence and significant pointing errors, where the parameters are ${\gamma }_{th,l}^{FSO}=3\mathrm{dB}$, ${\gamma }_{th,u}^{FSO}=7\mathrm{dB}$, and ${\gamma }_{th}^{FSO}={\gamma }_{th}^{THz}=4\mathrm{dB}$, we examine the system’s channel capacity with varying numbers of links (N = 5, 10, 15, 20). This analysis explores the capacity implications of employing IRS versus traditional configurations without IRS, across different link counts. Figure 10 illustrates that the system’s channel capacity decreases as the number of links increases. This degradation is attributed to the utilization of serial IRS-UAV configurations as relays; a higher count of serial relays elevates the error probability within any given relay, thereby diminishing the overall channel capacity. Furthermore, the figure indicates that systems incorporating IRS technology exhibit superior performance compared to those without. Specifically, when the IRS count is set to four and the number of links to ten, the system’s outage probability significantly outperforms scenarios where IRS is absent and the link count is four. This demonstrates that IRS deployment can effectively counteract the performance decline associated with an increased number of links.

5. Conclusions

To enhance rapid deployment of post-disaster communication networks, extend coverage, and address the challenges of future 6G communications and information warfare, this study introduces an innovative hybrid multi-hop FSO/THz link system for UAVs. The FSO and THz links are modeled following the gamma–gamma and respective distributions, providing a robust framework for analyzing system performance under varying conditions. This research incorporates the impact of atmospheric turbulence, pointing errors, and AOA fluctuations on both FSO and THz links, which are critical factors in real-world applications. Utilizing the Meijer-G function and the complementary error function (erfc), we have derived closed-form expressions for the outage probability, BER, and average channel capacity of FSO multi-hop links, THz multi-hop links, and multi-hop links employing both hard- and soft-switching strategies. These mathematical formulations allow for a comprehensive evaluation of the system’s performance and facilitate a detailed analysis of the strategic differences in link management. Our findings underscore the effectiveness of both hard- and soft-switching strategies in enhancing the performance of the UAV-based hybrid multi-hop FSO/THz link system. Notably, soft switching provides a marginal gain over hard switching by reducing the necessity for frequent transitions between links, thereby optimizing system efficiency and reducing potential disruptions in communication continuity. The integration of IRS significantly mitigates the adverse effects of atmospheric conditions and alignment errors. Importantly, system performance escalates with the increase in the number of IRS reflective elements, demonstrating the scalability and adaptability of this approach. Employing UAVs as relays transcends the limitations of traditional relay systems and notably enhances system performance, particularly when IRS technology is integrated onto the UAVs. This configuration not only stabilizes the communication links but also extends their operational range and reliability. The strategic use of THz links as a backup for wireless optical communications further bolsters the system robustness, ensuring stable connectivity even under challenging conditions. Numerical analysis confirms the superiority of the proposed hybrid link system over conventional systems, highlighting its potential in revolutionizing UAV communication frameworks. This research not only provides valuable theoretical insights for the optimal design of UAV communication systems but also pioneers a novel approach for advancing next-generation communications. Looking forward, these insights lay a solid foundation for future experimental validations and practical deployments, promising to significantly impact the development of resilient and efficient communication infrastructures in both civilian and military contexts.